Discretization of the Laplacian operator using a multitude of overlapping cartesian grids

Adaptive discretizations are important in many multiscale problems, where it is critical to reduce the computational time while achieving the same or greater accuracy in particular regions of the computational domain. Finite differences applied on Cartesian grids are of course a very simple numerical method for solving differential equations, but do not allow adaptive discretizations, forcing the user to refine the computational domain globally. Moreover, when the grid is not cartesian, the discretization of the differential operators in space must take into account the metrics, making grid transformations a bit annoying to handle. This talk presents a 2D adaptive finite-difference method to discretize the Laplacian operator on a computational domain made of multiple overlapping grids, defined by a generic quadrilateral. Adaptive discretizations are important in compressible/incompressible flow problems since it is often necessary to resolve details on multiple levels allowing large regions of space to be modeled using a reduced number of degrees of freedom (reducing the computational time). There are a wide variety of methods for adaptively discretizing space, but Cartesian grids have often outperformed them even at high resolutions due to their simple and accurate numerical stencils and their superior parallel performances. The Laplace operator is an essential building block of the Navier-Stokes equations, a model that governs fluid flows. In this talk will be presented a 2D finite-difference approach to solve a Laplacian operator, applying patches of overlapping grids where a more fined level is needed, leaving coarser meshes in the rest of the computational domain. These overlapping grids will have generic quadrilateral shapes. Specifically, the talk will cover the following topics: introduction to the finite difference methods, domain partitioning, solution approximation; overview of different types of meshes to represent in a discrete way the geometry involved in a problem, with a focus on the octree data structure, presenting PABLO and PABLitO. The first one is an external library used to manage each single grid's creation, load balancing and internal communications, while the second one is the Python API of that library written ad hoc for the project; presentation of the algorithm used to communicate data between meshes (being all of them unaware of each other's existence) using MPI inter-communicators and clarification of the monolithic approach applied building the final matrix for the system to solve, taking into account diagonal, restriction and prolongation blocks; presentation of some results; conclusions, references